N ni

=-i^T / dqjeKQ{qn,tn-,qj,tj)Hi(qj)KQ{qj,tj\qQ„tQ) j=i J

= -i J dq J dtK0(qn,tn;q,t)Hi(q)Ko{q,t;q0,t0) . (14.35)

Note that the first order term is obtained by retaining the first order contribution from exp(—ieH/) to only one time "slice" (if Hi contributes to two time slices, the result is already second order) and summing (integrating) over all possible times at which Hi can contribute. The time summation is converted to a time integral using e —» dt. Now, inserting this into the expression for Si and using M' = 1 give

(S0a)1 = -i J dqndq0dq £ dt4>*0(qn,tn)Ko(qn,tn\q,t)Hi(q)

= -idtJ dq<fy(q,t)Hi(q)<l>a(q,t) . (14.36)

However, since Hi = Hi(q) — 0(q,t\Hi(Q(t))\q',t)06(q - <?'), with the operator Q in the interaction representation (in agreement with the formalism of Chapter 3), this equation can be written

(S0a)1 = -ij^dt Jdqdq' (0\q, t)00(q, t\H,(Q(t))\q', t)00(q', i|a)

where, in the last step, the completeness of the position eigenfunctions at any time

las been used to remove the integrations over q and q'. Note that this result agrees with (14.28), proving the equality to first order in Hi. Note also that in this form Hi is now an operator, depending on the generalized coordinates Q.

Similarly, K2 can be obtained by using the above expansion at two different times, n—1j-1

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